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Local Systems in Groups LOCAL SYSTEMS IN GROUPS Dedicated to the memory of Hanna Neumann J. A. HULSE (Received 27 June 1972) Communicated by M. F. Newman 1. Introduction Throughout this paper we will assume that all groups are contained in some fixed but arbitrary universe. Thus the class £) of all groups becomes a set. If X is a class of groups then we assume that 1 e X and if H = Ge£ then H eX. German capitals will be used to denote classes of groups. We will use an extended form of the notation for closure operators on the set of all classes of groups introduced by Hall in [1]. If H is any set, let -^(H) be the set of all subsets of H. DEFINITION 1.1. If ^£ 0>{H) for some set H, a map A: & ->• & is called an operator on & if X c A(X)S A(Y) for all X, Ye&> with X c Y. If also a is an ordinal number the operators A11 and A* on t? are denned by A°(X) = X, A\X) = A(u {A\X)\ P < a}) if a > 0 and A*(X) = u {A*(X) | a is an ordinal} for all lef. A is called a closure operator if A = A*. A relation of partial order may be defined on the set of all such operators on 0> by the rule that A ^ B if and only of A(X) £ B(X) for all Xe0». Then clearly A1 ^ Aa+i ^ A* for all ordinals a and if, for some a, A" = A'+1 then A" = Afi = A* for all ft 2: a. From this we see that an operator on ^ is a closure operator if and only if A = A2. Further since 8P is a set it is clear that there exists an ordinal a such that A" = A*+ * = A*. Thus it follows that A* is a closure operator and is the least closure operator B such that A ^ B. We will use small capitals to denote operators on the set of all classes of groups (in a given universe) and Roman capitals to denote operators on ^(^(K)) for some set K. In this paper we are concerned with operators derived from various types of local systems. 426 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 02 Oct 2021 at 06:49:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700018061 [2] Local systems in groups 427 DEFINITION 1.2. A set £ of subsets of a set H is called a local system for H if 0 ¥= & and each finite subset of H is contained in some member of 2.. J> is called weakly upper-directed if for each Q e 1 and each h e H there exists ReJ such that Q u {h} £ R and 2. is called a fower if J2 is well-ordered by inclusion. DEFINITION 1.3. The operators L, LW, LU, L, are defined by GetX LUX, LfX, respectively) if and only if G has a local system J of ^-subgroups (which is weakly upper-directed, upper-directed, a tower, respectively) for all classes of groups X. The operators L,LW, Lu, Lt on ^(^(K)) are defined similarly with ^(^(K)) replacing the set of all classes of groups. We remark that in our definition of a local system we are following Hartley [2]. In [4] Kuros requires every local system to be upper-directed. It follows immediately from the definition, that (1) Lt ^ Lu ^ Lw ^ L and hence for all ordinals a a (2) h* g L"U ^ i£ S L and L? ^ L* ^ L£ g L* . It is well known that L = L* and LUX = L*X if X — sX, where GesX if and only if G is a subgroup of some 3t-group. In [3] Hickin showed that Lu < L*. The first aim of this paper is to show that the inequalities in (1) and (2) are all 1 1 strict, with the exception that Lt* = L* and also that A" < A"" " < A* in the cases where A = L,, LU and LW with some restriction on the cardinality, a |, of the ordinals a. These results are given in THEOREM A. Let cc be an ordinal number. Then: (i) i4 < L ; +1 (ii) if \a\ ^ Ko then L"W< L"W < L*; (iii) L* < L* and L" < L£; Xo +i (iv) (Hickin) if \a\ ^ 2 then L*u < L*a < L*; +1 (v) Lu $ t?, L^ < L^ and L* < L? < i*; (vi) Lt = C Thus, in particular, the operators Lt, LU and LW are not closure operators and Lt* = L* < L* < L* = L. In Section 2 of the paper we prove some purely set theoretic lemmas required for the proof of Theorem A while in Section 3 we develop the necessary translation into group theory. This investigation was sparked off by the question as to whether the class £s, the class of all groups G in which the set of all serial subgroups of G forms a complete sublattice of the lattice of all subgroups of G, is L-closed. This question arose from the work of Hartley in [2] where he showed that fis contains all Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 02 Oct 2021 at 06:49:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700018061 428 J. A. Hulse [3] locally finite groups. We have been unable to completely decide this question but we will prove the following results in Section 4. THEOREM B. The class fis is Lw*-closed and if X = sX s £s then LX S £S. We remark that no such local theorem can hold for the classes fin, fld or £a, consisting of all groups in which the set of all subnormal, ascendant or decendant subgroups, respectively, forms a complete sublattice of the lattice of all sub- groups, since there exist locally nilpotent groups in which the join of two sub- normal subgroups is neither ascendant nor descendant — see for example [7, Appendix D> Exercise 23]. If 3Jtvand 9K^ are the classes of groups with the minimum condition on all subgroups and all serial subgroups, respectively, then we will deduce the following extension of Theorem A of [2] as a corollary of Theorem B. V V THEOREM C. 2RS <= £s and L2R <= £s. It is perhaps worth noting in contrast that an example in [2] shows that there are groups with the maximum condition on all subgroups which do not lie in fis. 2. The set theory We begin this section with LEMMA 2.1. For each ordinal oc and each set K with \K\^ max{X0,|oc|} there exists a subset 2. of 0>(K) such that 0) 0el; (ii) L/(J2) = L/(J) for all /?; (iii) L/(J) cz LU"(J) = LU*(J2) if p < a. PROOF, (iii) is just the contents of Lemma 1 of [3], (ii) is easily seen to follow from the construction in [3] and (i) follows since LU(J u{0}) = LU(J) u {0} and Lw(i LEMMA 2.2. // K is any set with \K\ ^ Ko then there exists a subset M of @(K) such that 0ei and 5 r PROOF. Let J " be the set of all finite subsets of K. Then \& \ = \K\ since K 5 is infinite. We now partition K into sets KF, one for each F e J ", with | KF | = | K | and let 3. = {0} u{FuKF\Fe^}, so that 0ei£ &>{K). Suppose now that Q, Re£\{0} and 2 £ R. Then Q = £ U K£ and i? == F U Kr for some E, F e 3?, so K£ £ F u KF and so, Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 02 Oct 2021 at 06:49:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700018061 [4] Local systems in groups 429 since | F | < Ko ^ | KE \, KE n Kr # 0. Thus E = F and so Q = R. Hence it follows that any weakly upper-directed local system which is contained in St must have at most one member distinct from 0, so LW(J) = 2. and so Lw*(.2) = £. However, 1 is a local system for X and K £ J and so 2. c L(J), as required. LEMMA 2.3. // X is any set with \K.\ = Ka /or some non-limit ordinal a ^ 1 f/iew f/iere exists a set J ^ ^(X) such that 0ei and PROOF. We are indebted to the referee for the following proof which is much shorter than our original. By Theorem 1 on p. 451 of Sierpinski [7] there exists a set ^ £ &>(K) such that (1) |^|> Ka and \R\ = K, for all Ret% and (2) if S, Te@ and S ^ T then |S n T| < K^. Now clearly & also satisfies (3) if S, TUT2,-- are countably many distinct elements of 3& then S $ (J"= I Tn. For each xeK let 3tx = {Re&|xeR} and let R = {x eK | \@x\< KJ . Then | u {^x xe£} | ^ Ka and so by (1) we may replace K by K\R and M by ^ \ u {^ x e R) in order to assume (4) \Stx\ ^ Ka for all xeK. Let M = {(x,n)|x6K, n = 0,1,2,-}. Since j Af j = \K\ = Ka we may well-order M = {tp\P <ooa}.
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